Inharmonicity
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Inharmonicity
by Julie Delisle
Timbre Lingo | Timbre and Orchestration Writings
Inharmonicity is a feature of timbre that is related to the frequency spectrum of a given sound. It is measured through an analysis of the partials, or, more precisely, of their frequencies and the intervals between them. We first extract the fundamental frequency with the help of various algorithms, and then we compare the frequencies of the partials contained in the spectrum with those of a perfectly harmonic spectrum (where the partials are situated at frequencies that are integral multiples of the fundamental frequency). The further away the frequencies of the partials are from those of the harmonic spectrum, the higher the inharmonicity value will be. From a mathematical perspective, inharmonicity is the "weighted sum of deviation of each individual partial from harmonicity"[1]. In other words, the calculation "takes in account the amount of energy outside the ideal harmonic series."[2]
Practically, inharmonicity has to do with the physical properties of the musical instruments, such as the density and/or degree of tension of the vibrating material. Bowed string and wind instruments have a rather harmonic spectrum, which means that pitches can be clearly heard. For wind instruments, inharmonicity depends on the properties of the tube and the way it is tuned. Usually, wind instrument builders try to obtain a spectrum whose partials are rather close to standard harmonics. But in the case of the piano, the strings are known to be slightly inharmonic because of their density and rigidity. Then, many bells and percussion instruments from the idiophones family typically produce quite inharmonic sounds.
Finally, the use of sound synthesis opens up the possibilities for the exploration of sound inharmonicity. By combining several oscillators and using techniques such as frequency modulation (FM) synthesis with analog technology, for instance, or by defining mathematical functions that correspond to inharmonic spectra with digital technology, it is possible to create an almost infinite variety of inharmonic timbres.
References
[1] PEETERS, Geoffroy, et al. 2011. « The TimbreToolbox : extracting audio descriptors from musical signals ». Journal of the Acoustical Society of America 130 (5), p. 2909.
[2] LARTILLOT, Olivier. 2017. "MIRtoolbox 1.7 User's Manual", online at <https://www.jyu.fi/hytk/fi/laitokset/mutku/en/research/materials/mirtoolbox/manual1-7.pdf>, last consulted March 15, 2019.
